Average Error: 28.8 → 0.1
Time: 27.3s
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12657.23948929309 \lor \neg \left(x \le 10295.895657573294\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} - \left(1 + x\right) \cdot \frac{1}{x - 1}\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -12657.23948929309 or 10295.895657573294 < x

    1. Initial program 59.2

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
    2. Using strategy rm

    3. Applied div-inv59.4

      \[\leadsto \frac{x}{x + 1} - \color{blue}{\left(x + 1\right) \cdot \frac{1}{x - 1}}\]

    4. Taylor expanded around -inf 0.3

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)}\]

    5. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}}\]

    if -12657.23948929309 < x < 10295.895657573294

    1. Initial program 0.1

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
    2. Using strategy rm

    3. Applied div-inv0.1

      \[\leadsto \frac{x}{x + 1} - \color{blue}{\left(x + 1\right) \cdot \frac{1}{x - 1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12657.23948929309 \lor \neg \left(x \le 10295.895657573294\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} - \left(1 + x\right) \cdot \frac{1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))

Details

Time bar (total: 27.1s)Debug log

sample73.0ms

Algorithm
intervals

simplify14.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
14.0ms
(- (/ x (+ x 1)) (/ (+ x 1) (- x 1)))

prune4.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 28.6b

localize22.0ms

Local error

Found 3 expressions with local error:

3.5b
(- (/ x (+ x 1)) (/ (+ x 1) (- x 1)))
0.0b
(/ x (+ x 1))
0.0b
(/ (+ x 1) (- x 1))

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
17×add-sqr-sqrt
14×*-un-lft-identity
12×add-cube-cbrt
10×times-frac
6×associate-/r*
5×add-log-exp
5×add-exp-log
5×add-cbrt-cube
4×associate-/r/
3×associate-/l*
3×pow1
2×associate-/l/
2×difference-of-squares
2×flip--
2×div-inv
2×flip-+
2×flip3--
2×flip3-+
2×frac-2neg
2×clear-num
1×distribute-lft-out--
1×frac-sub
1×div-exp
1×diff-log
1×sub-neg
1×cbrt-undiv
Counts
3 → 62
Calls
3 calls:
Slowest
7.0ms
(- (/ x (+ x 1)) (/ (+ x 1) (- x 1)))
3.0ms
(/ (+ x 1) (- x 1))
0.0ms
(/ x (+ x 1))

series83.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
49.0ms
(- (/ x (+ x 1)) (/ (+ x 1) (- x 1)))
17.0ms
(/ (+ x 1) (- x 1))
16.0ms
(/ x (+ x 1))

simplify1.9s

Counts
51 → 71
Calls
51 calls:
Slowest
446.0ms
(- (* x (- x 1)) (* (+ x 1) (+ x 1)))
271.0ms
(/ (* (* (+ x 1) (+ x 1)) (+ x 1)) (* (* (- x 1) (- x 1)) (- x 1)))
211.0ms
(- (+ (* 3 (/ 1 (pow x 3))) (+ (/ 1 (pow x 2)) (* 3 (/ 1 x)))))
208.0ms
(- (+ (* 2 x) (+ (* 2 (pow x 2)) 1)))
199.0ms
(- (+ (* 3 (/ 1 (pow x 3))) (+ (/ 1 (pow x 2)) (* 3 (/ 1 x)))))

prune496.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

8.6b
(- (/ x (+ x 1)) (* (+ x 1) (/ 1 (- x 1))))
0.1b
(* (+ x 1) (/ 1 (- x 1)))
0.0b
(/ 1 (- x 1))
0.0b
(/ x (+ x 1))

rewrite27.0ms

Algorithm
rewrite-expression-head
Rules
9×add-exp-log
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
7×pow1
6×add-log-exp
6×frac-sub
6×associate-/r/
6×associate-*r*
6×add-cbrt-cube
6×associate-/r*
5×flip-+
5×flip3-+
4×associate-*l/
4×frac-times
3×flip--
3×div-inv
3×associate-*l*
3×flip3--
2×un-div-inv
2×associate-*r/
2×prod-exp
2×frac-2neg
2×clear-num
2×rec-exp
1×inv-pow
1×pow-flip
1×cbrt-unprod
1×*-commutative
1×pow-prod-down
1×diff-log
1×sub-neg
Counts
4 → 78
Calls
4 calls:
Slowest
12.0ms
(- (/ x (+ x 1)) (* (+ x 1) (/ 1 (- x 1))))
8.0ms
(* (+ x 1) (/ 1 (- x 1)))
2.0ms
(/ 1 (- x 1))
1.0ms
(/ x (+ x 1))

series73.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
34.0ms
(- (/ x (+ x 1)) (* (+ x 1) (/ 1 (- x 1))))
16.0ms
(/ x (+ x 1))
13.0ms
(* (+ x 1) (/ 1 (- x 1)))
11.0ms
(/ 1 (- x 1))

simplify5.6s

Counts
56 → 90
Calls
56 calls:
Slowest
549.0ms
(- (* x (- x 1)) (* (+ x 1) (+ x 1)))
522.0ms
(* (- (* x x) (* 1 1)) (/ 1 (- x 1)))
385.0ms
(* (+ x 1) (+ (* x x) (- (* 1 1) (* x 1))))
362.0ms
(/ (exp (/ x (+ x 1))) (exp (* (+ x 1) (/ 1 (- x 1)))))
347.0ms
(* (+ x 1) (* (- x 1) (- x 1)))

prune631.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

8.6b
(- (/ x (+ x 1)) (* (+ x 1) (/ 1 (- x 1))))
8.6b
(- (/ x (+ x 1)) (* (+ x 1) (/ 1 (- x 1))))
0.1b
(* (+ x 1) (/ 1 (- x 1)))
0.1b
(* (+ x 1) (/ 1 (- x 1)))

rewrite45.0ms

Algorithm
rewrite-expression-head
Rules
12×frac-sub
12×add-exp-log
12×associate-*r*
8×add-log-exp
8×flip-+
8×add-cube-cbrt
8×associate-*l/
8×add-cbrt-cube
8×flip3-+
8×*-un-lft-identity
8×pow1
8×frac-times
8×add-sqr-sqrt
6×associate-*l*
4×flip--
4×un-div-inv
4×associate-/r/
4×associate-*r/
4×prod-exp
4×flip3--
2×div-inv
2×cbrt-unprod
2×*-commutative
2×pow-prod-down
2×diff-log
2×sub-neg
2×rec-exp
Counts
4 → 90
Calls
4 calls:
Slowest
12.0ms
(- (/ x (+ x 1)) (* (+ x 1) (/ 1 (- x 1))))
12.0ms
(- (/ x (+ x 1)) (* (+ x 1) (/ 1 (- x 1))))
10.0ms
(* (+ x 1) (/ 1 (- x 1)))
8.0ms
(* (+ x 1) (/ 1 (- x 1)))

series121.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
51.0ms
(- (/ x (+ x 1)) (* (+ x 1) (/ 1 (- x 1))))
49.0ms
(- (/ x (+ x 1)) (* (+ x 1) (/ 1 (- x 1))))
11.0ms
(* (+ x 1) (/ 1 (- x 1)))
10.0ms
(* (+ x 1) (/ 1 (- x 1)))

simplify10.5s

Counts
78 → 102
Calls
78 calls:
Slowest
637.0ms
(* (- (* x x) (* 1 1)) (/ 1 (- x 1)))
612.0ms
(* (- (* x x) (* 1 1)) (/ 1 (- x 1)))
518.0ms
(- (* x (- x 1)) (* (+ x 1) (+ x 1)))
480.0ms
(- (* x (- x 1)) (* (+ x 1) (+ x 1)))
435.0ms
(* (+ x 1) (+ (* x x) (- (* 1 1) (* x 1))))

prune1.1s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize29.0ms

Local error

Found 4 expressions with local error:

5.6b
(+ (- (/ -3 x) (/ 1 (* x x))) (/ (/ -3 x) (* x x)))
0.3b
(/ 1 (* x x))
0.1b
(/ (/ -3 x) (* x x))
0.0b
(- (/ -3 x) (/ 1 (* x x)))

rewrite52.0ms

Algorithm
rewrite-expression-head
Rules
16×*-un-lft-identity
11×add-log-exp
8×add-sqr-sqrt
7×add-exp-log
6×div-inv
6×add-cube-cbrt
6×add-cbrt-cube
5×pow1
4×distribute-lft-out--
4×associate-/l*
4×times-frac
3×distribute-lft-out
3×frac-add
2×flip--
2×pow-flip
2×frac-sub
2×diff-log
2×flip3--
2×frac-2neg
2×sub-neg
2×sum-log
2×associate-/r*
2×clear-num
2×associate-+l+
1×associate-/l/
1×difference-of-squares
1×inv-pow
1×flip-+
1×associate-+l-
1×div-exp
1×flip3-+
1×rec-exp
1×+-commutative
1×cbrt-undiv
1×pow2
Counts
4 → 74
Calls
4 calls:
Slowest
37.0ms
(+ (- (/ -3 x) (/ 1 (* x x))) (/ (/ -3 x) (* x x)))
7.0ms
(- (/ -3 x) (/ 1 (* x x)))
5.0ms
(/ (/ -3 x) (* x x))
1.0ms
(/ 1 (* x x))

series438.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
377.0ms
(- (/ -3 x) (/ 1 (* x x)))
26.0ms
(/ (/ -3 x) (* x x))
22.0ms
(+ (- (/ -3 x) (/ 1 (* x x))) (/ (/ -3 x) (* x x)))
13.0ms
(/ 1 (* x x))

simplify2.9s

Counts
47 → 86
Calls
47 calls:
Slowest
568.0ms
(/ (* (* (/ -3 x) (/ -3 x)) (/ -3 x)) (* (* (* x x) (* x x)) (* x x)))
361.0ms
(* (+ (* (/ -3 x) (/ -3 x)) (+ (* (/ 1 (* x x)) (/ 1 (* x x))) (* (/ -3 x) (/ 1 (* x x))))) (* x x))
271.0ms
(+ (* (- (* (/ -3 x) (/ -3 x)) (* (/ 1 (* x x)) (/ 1 (* x x)))) (* x x)) (* (+ (/ -3 x) (/ 1 (* x x))) (/ -3 x)))
228.0ms
(- (+ (* 3 (/ 1 (pow x 3))) (+ (/ 1 (pow x 2)) (* 3 (/ 1 x)))))
221.0ms
(- (+ (* 3 (/ 1 (pow x 3))) (+ (/ 1 (pow x 2)) (* 3 (/ 1 x)))))

prune623.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes44.0ms

Accuracy

100% (0.0b remaining)

Error of 0.1b against oracle of 0.1b and baseline of 28.9b

bsearch167.0ms

end0.0ms

sample2.1s

Algorithm
intervals